Abelian variety isogeny class 2.61.az_kk over $\F_{61}$

• Label: 2.61.az_kk
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 - 25 x + 270 x^{2} - 1525 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.0568212926978$, $\pm0.288610051981$
Angle rank:	$2$ (numerical)
Number field:	4.0.1267596.1
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2442$	$13533564$	$51531464952$	$191720581827456$	$713320847896463202$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3637$	$227032$	$13846801$	$844570177$	$51519854842$	$3142738178197$	$191707293412993$	$11694146165782792$	$713342913604307677$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=6 x^6+58 x^5+15 x^4+4 x^3+52 x^2+48 x+24$
• $y^2=8 x^6+46 x^5+50 x^4+39 x^3+31 x^2+29 x+51$
• $y^2=7 x^6+42 x^5+58 x^4+7 x^3+35 x^2+43 x+42$
• $y^2=43 x^6+58 x^5+18 x^4+18 x^3+47 x^2+47 x+7$
• $y^2=38 x^6+22 x^5+5 x^4+12 x^3+13 x^2+35 x+49$
• $y^2=4 x^6+20 x^5+52 x^4+3 x^3+37 x^2+31 x+41$
• $y^2=5 x^6+53 x^5+13 x^4+31 x^3+34 x^2+34 x+48$
• $y^2=8 x^6+3 x^5+31 x^4+59 x^3+13 x^2+8 x+20$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is 4.0.1267596.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.61.z_kk	$2$	(not in LMFDB)